Momentum operator

The momentum operator in quantum mechanics is the operator for momentum measurement of particles. The momentum operator in one dimension is in the coordinate representation is given by:

It denotes the imaginary unit, the reduced Planck's quantum of action and the partial derivative in the direction of the spatial coordinate. The nabla operator is obtained in three dimensions the vector

The physical state of a particle is given mathematically in quantum mechanics by an associated vector of a Hilbert space. This state is thus described in the Bra- Ket notation by the vector. The observables are represented by self-adjoint operators on. Specifically, the momentum operator is the summary of the three observables, so that

The average value ( expected value ) of the measurement results of the j-th component of the momentum of the particle in the state.

Definition and properties

• In the canonical quantization we interpret the phase space coordinates, the location and the momentum of the classical system, as self-adjoint operators of a Hilbert space and calls for the local operators and momentum operators, the canonical commutation

• From the canonical commutation relations it follows that the three components of the pulse are jointly measurable and that its spectrum consists ( range of possible values) of the entire room. The possible pulses are therefore not quantized, but continuously.
• The spatial representation is defined by the spectral representation of the position operator. The Hilbert space is the space of square integrable complex functions of the local space, each state is given by a spatial wave function. The local operators, the operators of the coordinates multiplication functions, that is, the local operator acts on local wave functions by multiplying the wave function of the coordinate function

• In the momentum representation of the momentum operator has a multiplicative effect on square integrable pulse wave functions

• The position and momentum operators are linear combinations of creation and annihilation operators:

Why is the momentum operator in coordinate representation is a differential operator?

According to the Noether theorem belongs to every continuous symmetry of the action of a conserved quantity, and vice versa. For example, the pulse is exactly obtained even when the effect is translationally invariant. In the Hamiltonian formulation, the conserved quantity generates the symmetry transformation in the phase space by their Poisson bracket, the pulse generates shifts.

When applied to a wave function gives each shift by the shifted function that has the value at each point, the archetype had on,

The generatrix of these parameter family of shifts thus defined up to a factor of the pulse, ie the pulse satisfies by definition

The factor dimensional reasons occurs because the product of momentum and place has the dimension of angular momentum or an effect. The imaginary unit is required since a unitary transformation, and the pulse is to be self-adjoint. Directs the equation

Off to on, the result is the momentum operator as a derivative with respect to the place

The fact that the momentum operator in position space take the form, can be even without the knowledge of the associated unitary operator as follows deduced from the Noether theorem: One reconstructed first from the Schrödinger equation, the associated Lagrangian and then determine explicitly the expected value obtained by an infinitesimal displacement of the wave function.